Nuprl Lemma : free-group-adjunction
FreeGp -| ForgetGroup
Proof
Definitions occuring in Statement :
counit-unit-adjunction: F -| G,
forget-group: ForgetGroup,
free-group-functor: FreeGp,
group-cat: Group,
type-cat: TypeCat
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
forget-group: ForgetGroup,
free-group-functor: FreeGp,
group-cat: Group,
all: ∀x:A. B[x],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_apply: x[s],
mk-cat: mk-cat,
subtype_rel: A ⊆r B,
cat-ob: cat-ob(C),
pi1: fst(t),
guard: {T},
uimplies: b supposing a,
grp: Group{i},
type-cat: TypeCat,
free-group: free-group(X),
grp_car: |g|,
counit-unit-equations: counit-unit-equations(D;C;F;G;eps;eta),
and: P ∧ Q,
cand: A c∧ B,
mon: Mon,
compose: f o g,
fg-lift: fg-lift(G;f),
fg-hom: fg-hom(G;f;w),
list_accum: list_accum,
free-letter: free-letter(x),
cons: [a / b],
nil: [],
it: ⋅,
infix_ap: x f y,
grp_op: *,
pi2: snd(t),
free-append: w + w',
append: as @ bs,
list_ind: list_ind,
grp_id: e,
free-0: 0,
free-word: free-word(X),
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
imon: IMonoid,
prop: ℙ,
monoid_hom: MonHom(M1,M2),
grp_inv: ~,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
mk-adjunction_wf,
type-cat_wf,
group-cat_wf,
free-group-functor_wf,
forget-group_wf,
ob_mk_functor_lemma,
istype-void,
cat_arrow_triple_lemma,
fg-lift_wf,
grp_car_wf,
mon_subtype_grp_sig,
grp_subtype_mon,
subtype_rel_transitivity,
subtype_rel_self,
grp_wf,
cat-ob_wf,
free-letter_wf,
cat_ob_pair_lemma,
cat_comp_tuple_lemma,
arrow_mk_functor_lemma,
cat_id_tuple_lemma,
istype-universe,
monoid_hom_wf,
free-group-generators,
free-group_wf,
compose_wf_for_mon_hom,
id-is-monoid_hom,
free-word_wf,
list_accum_cons_lemma,
list_accum_nil_lemma,
mon_ident,
grp_sig_wf,
monoid_p_wf,
grp_op_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
list_ind_nil_lemma,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
hypothesisEquality,
applyEquality,
because_Cache,
independent_isectElimination,
universeIsType,
universeEquality,
lambdaFormation_alt,
independent_pairFormation,
setElimination,
rename,
inhabitedIsType,
functionExtensionality_alt,
functionIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
functionExtensionality,
setIsType,
productElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
FreeGp -| ForgetGroup
Date html generated:
2019_10_31-AM-07_25_15
Last ObjectModification:
2018_11_08-PM-06_01_02
Theory : small!categories
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